Let \(\sigma (A)={{\mathbb {C}}}\setminus \rho (A)\)  denote the spectrum of \(A\) . It can be used for the following
characterisation of the holomorphic semigroup generators, [1]}, Chapter IX:

Operator \(A\)  in a Banach space \(\mathfrak {X}\)  is the generator of a bounded holomorphic semigroup
of semi-angle \(\theta \in (0, \pi /2]\)  if and only if \(A\)  is a closed operator with a dense domain
\({\rm {dom}}(A)\)  such that:
\(\sigma (A) \subseteq \left\lbrace  z \in {{\mathbb {C}}},\ |\arg (z)|\le {\pi \over 2}-\theta \right\rbrace  \ , \quad 0<\theta \le {\frac{\pi }{2}} \ ,\) 
